发表机构
Inria; CNRS; Centrale Lille（法国国家信息与自动化研究所; 法国国家科学研究中心; 里尔中央理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出结构化马尔可夫决策过程中最优策略的几何理论，证明策略重构复杂度由决策边界几何而非状态空间大小决定，并给出统计保证与实验验证。
AI 中文摘要
经典动态规划通过值函数和策略来表示最优序贯决策。虽然这种函数表示对于计算最优决策是自然的，但它并不能直接识别出在最优策略固定后，支配策略重构、表示复杂度或预言机查询复杂度的数学对象。本文通过发展结构化最优策略的几何理论来回答这一问题，在该理论中，由策略引起的决策边界几何成为分析的主要对象。我们证明，在适当的结构正则性条件下，该几何提供了策略重构所需的最小表示，并决定了重构问题的统计和计算复杂度。基于这一表示，我们建立了策略诱导决策几何的结构性质，引入了边界和决策复杂度的内在概念，推导了决策压缩的信息论度量，并获得了从黑箱策略查询中进行边界估计和策略重构的统计保证。总的来说，这些结果表明，对于本文所考虑的结构化决策问题，策略重构的复杂度由决策边界的几何决定，而非由环境状态空间的基数决定。受控数值实验检验了主要理论预测，并提供了与所提出框架一致的实证证据。
英文摘要
Classical dynamic programming represents optimal sequential decisions through value functions and policies. While this functional representation is natural for computing optimal decisions, it does not directly identify the mathematical object governing policy reconstruction, representation complexity, or oracle-query complexity once an optimal policy is fixed. This paper addresses this question by developing a geometric theory of structured optimal policies in which the decision-boundary geometry induced by the policy becomes the primary object of analysis. We show that, under suitable structural regularity conditions, this geometry provides the minimal representation required for policy reconstruction and determines the statistical and computational complexity of the reconstruction problem. Building upon this representation, we establish structural properties of policy-induced decision geometry, introduce intrinsic notions of boundary and decision complexity, derive information-theoretic measures of decision compression, and obtain statistical guarantees for boundary estimation and policy reconstruction from black-box policy queries. Collectively, these results demonstrate that, for the structured decision problems considered here, the complexity of policy reconstruction is governed by the geometry of the decision boundary rather than by the cardinality of the ambient state space. Controlled numerical experiments examine the principal theoretical predictions and provide empirical evidence consistent with the proposed framework.
